Question 1

Given the rational function $$R(x) = \frac{2x}{x^2$-4}$$, which of the following statements is TRUE about its vertical asymptotes?

Accepted Answer

There are vertical asymptotes at $$x = 2$$ and $$x = -2.$$

Question 2

For the function $$f(x) = \frac{x^2$-1}{x}$$, what is the $$y-$$intercept?

Accepted Answer

$$y = -1$$

Question 3

Which of the following is the correct horizontal asymptote for $$f(x) = \frac{3x^2$-3x}{x^2$+x-12}$$?

Accepted Answer

$$y = 3$$

Question 4

A company manufactures cylindrical cans with a fixed volume of $$500 cm$^3. $$The top and bottom cost $$0.06$$ per cm$$^2$$ and the side costs $$0.03$$ per cm$$^2. $$Which equation best represents the total cost $$C(r) as a $$function of the radius $$r$$?

Accepted Answer

Question 5

Given $$f(x) = \frac{x^2$-1}{x}$$, which of the following describes the end behavior as $$x 	o \infty$$?

Accepted Answer

$$f(x) 	o 1$$

Question 6

Which of the following rational functions has a hole at $$x = 2$$?

Accepted Answer

$$f(x) = \frac{(x-2)(x+1)}{(x-2)(x-3)}$$

Question 7

A box with a square base and no top must have a volume of $$10,000$$ cubic inches. If $$x is $$the length of a side of the base, which function gives the surface area $$S(x) of $$the box?

Accepted Answer

$$S(x) = x^2$ + 4\frac{10,000}{x}$$

Question 8

Which of the following is the correct process to find the $$x-$$intercepts of a rational function $$R(x) = \frac{P(x)}{Q(x)}$$?

Accepted Answer

Set $$P(x) = 0$$ and solve for $$x.$$

Question 9

For the function $$f(x) = \frac{x^2$-1}{x}$$, which of the following describes all vertical asymptotes?

Accepted Answer

$$x = 0$$

Question 10

A rational function has a horizontal asymptote at $$y = 2. $$Which of the following could be its equation?

Accepted Answer

$$f(x) = \frac{2x^2$+1}{x^2$-3}$$

Question 11

If a rational function $$f(x)$$ has a degree of numerator less than the degree of denominator, what is the horizontal asymptote?

Accepted Answer

$$y = 0$$

Question 12

Given $$f(x) = \frac{x^2$-4}{x-2}$$, what is the value of $$f(2)$$ after simplifying the function?

Accepted Answer

$$f(2) is $$undefined

Question 13

A rational function $$f(x)$$ has a vertical asymptote at $$x = 5$$ and a horizontal asymptote at $$y = 3. $$Which of the following could be $$f(x)$$?

Accepted Answer

$$f(x) = \frac{3x}{x-5}$$

Question 14

Which of the following best describes the steps to analyze a rational function $$R(x) = \frac{P(x)}{Q(x)}$$?

Accepted Answer

Write in lowest terms, find $$x- $$and $$y-$$intercepts, locate vertical and horizontal asymptotes, analyze end behavior, and sketch the graph.

Question 15

If $$f(x) = \frac{x^2$-1}{x}$$, what is the domain of $$f(x)$$?

Accepted Answer

All real numbers except $$x = 0$$

Question 16

A rational function $$f(x)$$ has a removable discontinuity (hole) at $$x = 3. $$Which of the following must be true?

Accepted Answer

Both the numerator and denominator have a factor of $$(x-3).$$

Question 17

For the function $$f(x) = \frac{2x^2$+3x-2}{x^2$-1}$$, what are the $$x-$$intercepts?

Accepted Answer

$$x = -2, \frac{1}{2}$$

Question 18

A rational function $$f(x)$$ has a slant (oblique) asymptote. Which of the following must be true about the degrees of the numerator and denominator?

Accepted Answer

The degree of the numerator is exactly one more than the degree of the denominator.

Question 19

Suppose you are designing a can with a fixed volume and want to minimize the cost. Which mathematical process should you use?

Accepted Answer

Express cost as a function of one variable, then use calculus or graphing to find the minimum.

Question 20

If $$f(x) = \frac{x^2$-4}{x^2$-1}$$, what are the vertical asymptotes?

Accepted Answer

$$x = 1$$ and $$x = -1$$

3.4 Analyzing and Graphing Rational Functions in Precalculus

Study Guide - Practice Questions

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